Or, just because you add fixed effects and blow up the standard error of a variable's coefficient doesn't mean that the variable in question is not substantively important.
For a case in point consider the article, "Poverty and Civil War: Revisiting the Evidence" by Djankow and Reynal-Querol. They argue that poverty does not cause conflicts as many have previously reported and they show that when you add country fixed effects to their model, the coefficient on lagged income in the civil war regression becomes insignificant. It is interesting to note that the coefficients themselves barely change when the estimation changes from OLS with time dummies to full two way fixed effects.
For example, the first two columns of their Table 1 (p. 17 in the link above) they report a coefficient for income on the onset of civil war in two otherwise identical models, one pooled OLS and the other Fixed Effects. The OLS model has a coefficient on lagged income of -.08 with a t-stat of 4.16, while the corresponding coefficient in the FE model is -.09 with a t-stat of 1.74.
The FE coefficient is actually bigger, but its standard error is about 2.7 times larger (.0192 for OLS vs .052 for FE) so it becomes "insignificant".
Income is fairly persistent (autocorrelated) while civil wars are not, so taking all cross country information out of the mix is going to inflate the standard errors on the persistent variables (by reducing the variance of the independent variable). Lant Pritchett makes this point in the context of growth regressions in his 2000 paper in the World Bank Economic Review 2000, p. 239 "Understanding Patterns of Economic Growth" and also describes how using FE can increase problems resulting from measurement error.
In sum, it's one thing show that the effect of a variable in a panel comes totally from cross sectional differences and thus is suspect because it may be correlated with non-observed country specific factors. This would imply that putting in FE should push the coefficient on the offending variable toward zero. However, when the coefficient stays the same but becomes insignificant, especially when we know the variable is quite persistent over time, it is far from clear that the effect is really not there. It is likely that you've just inflated the coefficient variance enough to push the coefficient to statistical insignificance, even though its size is unchanged from the OLS regression.
Fixed effects is not always (dare I even say not often) the best estimator in panel models when we care about the economic effects of persistent independent variables.